Physics • Year 12 • Module 8 • Lesson 3
Hubble's Law and the Expanding Universe
Apply Hubble's law, redshift calculations, and the scale factor to real spectral data, graphical analysis and multi-step quantitative problems.
1. Interpret spectral data — galaxy recession velocities
Five galaxies have been observed with spectrographs. The table records the rest wavelength and observed wavelength of their Hα emission line (rest λrest = 656.3 nm). Use H0 = 70 km/s/Mpc. 10 marks
| Galaxy | λobs (nm) | Redshift z | Recession velocity v (km/s) | Distance d (Mpc) |
|---|---|---|---|---|
| NGC-A | 659.2 | |||
| NGC-B | 669.4 | |||
| NGC-C | 689.1 | |||
| NGC-D | 722.0 | |||
| NGC-E | 787.6 |
1.1 Complete all three calculated columns in the table above. Show your working for NGC-C below. 6 marks (1 per complete row)
1.2 Galaxy NGC-E has a relatively large redshift. State one reason why using v = cz becomes less reliable as z increases. 2 marks
1.3 Using your calculated distances, identify which galaxy (NGC-A through NGC-E) would appear as the faintest in the night sky, assuming all five galaxies have the same intrinsic luminosity. Justify your answer. 2 marks
2. Interpret a Hubble diagram
The graph below is a simplified Hubble diagram plotting recession velocity (km/s) against distance (Mpc) for a set of Type Ia supernovae. Use H0 = 70 km/s/Mpc. 7 marks
Figure 2.1. Illustrative Hubble diagram for a set of Type Ia supernovae used as standard candles. H0 = 70 km/s/Mpc.
2.1 Describe the relationship shown in the Hubble diagram and identify what the gradient of the best-fit line represents. 2 marks
2.2 Use the graph to estimate the recession velocity of a galaxy at 350 Mpc. Show how you would check your reading using Hubble's law. 2 marks
2.3 The data points do not lie exactly on the best-fit line. State two physical reasons (not experimental error) why individual galaxies might deviate from the ideal Hubble relationship. 3 marks
3. Predict and justify — scale factor and cosmic history
The James Webb Space Telescope has detected a galaxy at redshift z = 12. 5 marks
3.1 Calculate the scale factor of the universe when the light from this galaxy was emitted. Show your working. 2 marks
3.2 Interpret what this scale factor means for the size of the universe at that time compared with today. 1 mark
3.3 Using the low-z approximation, estimate the lookback time for this galaxy in Gyr (use H0 = 70 km/s/Mpc; 1 Mpc = 3.086 × 1019 km). Comment on whether the low-z approximation is appropriate here and why. 2 marks
4. Compare — redshift, recession velocity and distance for four objects
Complete the two-column comparison for each pair of objects. Use H0 = 70 km/s/Mpc and c = 3.00 × 105 km/s. 8 marks (1 per correct cell)
| Feature | Galaxy P (z = 0.03) | Galaxy Q (z = 0.15) |
|---|---|---|
| Recession velocity (km/s) | ||
| Distance (Mpc) | ||
| Scale factor at emission | ||
| Observed Hα wavelength (nm) given λrest = 656.3 nm |
Q1.1 — Galaxy data table
Using z = (λobs − 656.3) / 656.3, v = cz (c = 3.00 × 105 km/s), d = v / 70:
NGC-A: z = (659.2 − 656.3)/656.3 = 0.00442; v = 1326 km/s; d = 18.9 Mpc.
NGC-B: z = (669.4 − 656.3)/656.3 = 0.0200; v = 5994 km/s; d = 85.6 Mpc.
NGC-C: z = (689.1 − 656.3)/656.3 = 0.0500; v = 14 990 km/s; d = 214 Mpc. (Working: z = 32.8/656.3 = 0.0500; v = 0.0500 × 300 000 = 15 000 km/s; d = 15 000/70 = 214 Mpc.)
NGC-D: z = (722.0 − 656.3)/656.3 = 0.1001; v = 30 025 km/s; d = 429 Mpc.
NGC-E: z = (787.6 − 656.3)/656.3 = 0.200; v = 60 000 km/s; d = 857 Mpc.
Q1.2 — Reliability of v = cz for large z
The relation v = cz is a non-relativistic, linear approximation valid only for z < ~0.1. For larger redshifts the special relativistic Doppler formula (or full cosmological treatment) must be used, otherwise recession velocity is overestimated. At z = 0.20, the low-z formula gives v = 0.20c, but the relativistic formula gives a slightly different value because time dilation and length contraction become non-negligible.
Q1.3 — Faintest galaxy
NGC-E is faintest because it is the most distant (~857 Mpc). Apparent brightness decreases with the inverse square of distance, so for equal intrinsic luminosity, the most distant galaxy will appear faintest.
Q2.1 — Hubble diagram description and gradient
The graph shows a linear (straight-line) relationship between recession velocity and distance, passing through the origin. The gradient of the best-fit line equals the Hubble constant H0 (units km/s/Mpc), the constant of proportionality in Hubble's law.
Q2.2 — Reading from graph
Reading from the graph at d = 350 Mpc: v ≈ 24 500 km/s. Check: v = H0d = 70 × 350 = 24 500 km/s. Consistent.
Q2.3 — Physical reasons for scatter
Any two of: (1) Peculiar velocities — galaxies have their own random motions relative to the overall Hubble flow (gravitational attraction to nearby galaxy clusters), which adds or subtracts a few hundred km/s to the measured recession velocity. (2) Gravitational interactions — galaxies in dense clusters are gravitationally bound and may not follow pure Hubble expansion. (3) Non-uniform matter distribution (large-scale structure) — voids and filaments cause slight deviations from smooth expansion in different directions.
Q3.1 — Scale factor at z = 12
1 + z = 1 / athen, so athen = 1 / (1 + 12) = 1/13 ≈ 0.077.
Q3.2 — Interpretation of scale factor
When this light was emitted, the universe was approximately 1/13 (about 7.7%) of its current size — far smaller and denser than today.
Q3.3 — Lookback time and validity of approximation
Low-z approximation: t ≈ z / H0. H0 in s−1: 70 km/s/Mpc = 70 × 103 / (3.086 × 1022) = 2.27 × 10−18 s−1. t = 12 / (2.27 × 10−18) = 5.29 × 1018 s ≈ 168 Gyr. This is clearly unphysical (much greater than the age of the universe at ~13.8 Gyr), demonstrating that the low-z approximation is completely invalid for z = 12. Full numerical integration of the Friedmann equations gives a lookback time of ~13.4 Gyr for this galaxy (near the beginning of the universe). Only use t ≈ z/H0 for z < ~0.1.
Q4 — Comparison table
Galaxy P (z = 0.03): v = 0.03 × 3 × 105 = 9 000 km/s. d = 9 000 / 70 = 128.6 Mpc. athen = 1/(1 + 0.03) = 0.971. λobs = 656.3 × (1 + 0.03) = 676.0 nm.
Galaxy Q (z = 0.15): v = 0.15 × 3 × 105 = 45 000 km/s. d = 45 000 / 70 = 642.9 Mpc. athen = 1/(1 + 0.15) = 0.870. λobs = 656.3 × 1.15 = 754.7 nm.